# Automorphic Numbers with 10 Digits

## Theorem

The only $10$-digit automorphic numbers are:

$1 \, 787 \, 109 \, 376$
$8 \, 212 \, 890 \, 625$

## Proof

We have:

 $\displaystyle 1 \, 787 \, 109 \, 376^2$ $=$ $\displaystyle 3 \, 193 \, 759 \, 92 \, \mathbf {1 \, 787 \, 109 \, 376}$ $\quad$ $\quad$ $\displaystyle 8 \, 212 \, 890 \, 625^2$ $=$ $\displaystyle 67 \, 451 \, 572 \, 41 \mathbf {8 \, 212 \, 890 \, 625}$ $\quad$ $\quad$

thus demonstrating they are automorphic.